Generalizing the bottleneck matrix

Abstract

Given the Laplacian matrix associated to a weighted graph and given x a single vertex of it, the bottleneck matrix (related to x) is the inverse matrix of the sub matrix of the Laplacian obtained by eliminating the row and the column corresponding to x. The bottleneck matrix is used to calculate the group inverse of the initial Laplacian matrix, for instance. In this work we have managed to generalize this situation twofold: in the sense of considering symmetric M–matrices related to Schr¨odinger operators acting on networks (doubly weighted graphs, where not only edges but also vertices are discriminated) and also by using sub-matrices of the initial one in which two, three or more rows and columns are erased, those corresponding to two, three or more vertices. We conceive that every symmetric M–matrix corresponds to a network where both a conductance on the edges and a weight on the vertices are introduced. Solving boundary value problems for Schr¨odinger’s operators throughout the whole network or just a part of it, we find the relation between the corresponding group inverse and inverse matrices respectively. Since the part of the network to be considered is arbitrary, the reduction in the order of the matrices is also arbitrary. The work is finished by exposing the application of our result to the calculation of the Green function of a path.